Optimal Pricing for Multiple Services in Telecommunications Networks Offering Quality of Service Guarantees

Transcription

1 To Appear in IEEE/ACM Transactions on etworking, February 2003 Optimal Pricing for Multiple Serices in Telecommunications etworks Offering Quality of Serice Guarantees eil J. Keon Member, IEEE, G. Anandalingam, Senior Member, IEEE Abstract We consider pricing for multiple serices offered oer a single telecommunications network. Each serice has quality of serice (QoS) requirements that are guaranteed to users. Serice classes may be defined by the type of serice, such as oice, ideo or data, as well as the origin and destination of the connection proided to the user. We formulate the optimal pricing problem as a nonlinear integer epected reenue optimization problem. We simultaneously sole for prices and the resource allocations necessary to proide connections with guaranteed QoS. We derie optimality conditions and a solution method for this class of problems, and apply to a realistic model of a multi-serice communications network. Inde Terms Economics, etwork Design, Pricing, Quality of Serice (QoS) I. ITRODUCTIO n this paper, we derie a nonlinear mathematical I programming model for determining optimal prices for multiple serices with guaranteed quality of serice. We sole this model using an auction algorithm. The traditional distinction between oice networks, data networks and cable TV networks is fast becoming obsolete. In the future, multiple communications serices will be aailable to users oer a single network. These different serices will ary in terms of bandwidth requirements and tolerate different quality limitations such as loss of data and delay in transmission. Our model presents a way for network proiders to set prices for these serices, and allocate resources such that these Quality of Serice (QoS) requirements are guaranteed while epected reenues are maimized. A. Oeriew We consider the problem of using pricing and resource allocation to manage multiple serices networks with Qualityof-Serice guarantees. Although the arrial rate of connection Manuscript receied February 8, 200. This research was partially funded by a grant from the ational Science Foundation CR eil J. Keon is with the Co School of Business, Southern Methodist Uniersity, Dallas TX USA (phone: ; fa: ; nkeon@mail.co.smu.edu). G. Anandalingam is with R.H. Smith School of Business and the Institute for Systems Research, Uniersity of Maryland, College Park MD USA. ( ganand@rhsmith.umd.edu). requests can be fied by setting the price, the obsered arrials, and thus obsered reenue, are assumed to be a stochastic process. Also, the model we propose in this paper separates demand into classes, defined by serice type as well as origin and destination. Our model maimizes epected reenue in order to obtain optimal prices and the resources needed to guarantee QoS for each serice class. The resources we consider are buffer space for temporarily storing users data traffic, and bandwidth for transmitting the data through the network. All connections in a particular serice class (eg. oice, data, ideo etc.) share the same buffer and bandwidth allocation. The resource allocations must be sufficient to satisfy two QoS guarantees which are defined for eery serice class indiidually. The first QoS guarantee is a restriction on the probability of data being lost in the network. Data is lost when resources allocated to a serice class based on epected traffic calculations are insufficient to either transmit (using bandwidth) or store (using buffer space) all the data that is actually sent to the network by all users in a serice class. The second QoS guarantee is a limit on the delay, or time the data is buffered, during transmission by the network. The two QoS criteria constrain the problem in terms of buffer space and bandwidth that must be allocated to each serice class for a gien leel of demand. The trade-off between resource allocations among arious serice classes is complicated due to the fact that as the number of connections in a class increases we obsere economies of scale in bandwidth allocations. This is due to two phenomena: smoothing of data traffic in buffers and statistical multipleing gains. Economies-of-scale appear by decreasing the marginal resource allocations to any serice class, i.e. by decreasing the resources required to supply one additional connection. We construct a distributed search method called the auction algorithm to optimize the reenue by choosing prices, subect to constraints on the resource allocations implied by finite resources and QoS guarantees. Our search method takes adantage of seeral optimality properties we show for the resource allocations to each class. Using these properties we reduce the search space for the oerall problem and calculate optimal solutions.

2 To Appear in IEEE/ACM Transactions on etworking, February B. Related Literature We will briefly classify the related literature into two broad categories. First, we will discuss the bulk of the literature, which deals with pricing in best effort networks, such as the Internet. There has also been some recent work related to pricing connections for networks with QoS guarantees. ) Pricing for Best Effort Serice The most celebrated packet-switched network, the Internet, offers best effort serice, which is prone to unpredictable congestion and delays by definition. The current flow control scheme in the internet is called transmission control protocol (TCP) [2] [34]. Modifications to this scheme hae been suggested that would allow multiple serice classes, and induce users to behae fairly and efficiently, through simple or randomized packet marking mechanisms in the network [0][3] [4] [20]. Usage based schemes, which charge based on the actual resources used hae also been proposed for these types of networks [8] [29]. Priority pricing is another suggestion for allowing multiple serices oer best effort networks. The most well known work discusses a second bid auction, whereby it is incentie compatible, or in the users best interests to truthfully reeal their true aluation of serice in terms of a priority [22][23][24]. Another bidding paradigm, which results in a ash equilibrium among users was suggested for the Internet in [5]. A similar approach, again requiring users to bid for serice, was proposed for aailable bit rate serice, the besteffort serice offered in asynchronous transfer mode (ATM) networks [6]. A recent game theoretic model requiring users to choose from among seeral routes, each with its own delay, has been shown to hae a stable equilibrium solution where the relatie prices induce the desired operating point of the network operator [6][7]. An interesting problem of regulating the arrial of obs presented to the network is discussed in [26], and the informational requirements are relaed, using an adaptie on-line method in [25]. Finally, recent results hae been offered, based on dynamic programming that suggest a static price schedule results in network performance that is near optimal compared to congestion dependent pricing [27]. 2) Pricing with QoS Guarantees Typically a network with guaranteed QoS, such as the current oice network, must employ a call admission policy in order to satisfy the guarantees to users. In an alternatie approach to call admission, it has been suggested that users guarantee their own QoS by purchasing the required bandwidth and buffer resources for their desired QoS directly from the network [9] [2]. In [3], an analysis of a marketbased methodology is offered as eidence that pricing schemes can offer efficient resource allocations in connectionoriented networks offering QoS guarantees. (See also [33]) These models feature a conentional iew of pricing per connection and announcing the price to the public. The paper by de Veciana et al. [32] combines the conentional practice of prices being fied and announced, with the alternatie iew of allowing users to guarantee their own QoS by purchasing bandwidth and buffers directly. Resources are shared among users, incorporating multipleing gains. On-line negotiation is also suggested as a framework for connection set-up and allocation of network resources, using effectie bandwidth as a base for pricing in [] [2]. In a related paper that deals with single link point-to-point ATM networks, the pricing problem was formulated as a constrained optimal control problem, and soled using a three-stage solution procedure [35]. The limitation of the approach is computational tractability. Our paper belongs to this class of work, but uses a noel optimization model based on nonlinear mathematical programming. We use an optimal pricing scheme to moderate the demand for connection requests from different classes of serice. Each class of serice is defined by both the type of serice (eg. Voice, data, ideo), and by where the telecommunications flow originates, and where it is destined to. Thus, our paper tries to take into account the use of resources in the entire network. Based on the connection requests for each class of serice at the specified prices, the model estimates optimal buffer and bandwidth allocations to satisfy QoS requirements. Another contribution of this paper is the algorithm we propose to obtain the optimal solutions. C. Organization of the Paper This paper is organized as follows: In Section II, we will describe the etwork Model that we are considering, and eamine three important features, call blocking, loss probability, and delay in some detail. Section III contains the complete formulation of the epected reenue maimization model, and presents optimality properties. Section IV proides details of the Auction Algorithm used to sole the problem. We present the implementation of the algorithm to an etensie eample in Section V. We end the paper with concluding remarks in Section VI. II. A MODEL FOR OPTIMAL PRICIG WITH QOS GUARATEES In this section, we will gie an oeriew of the model that we will be using to derie optimal prices for the multiple serices in a telecommunications network where quality of serice (QoS) is guaranteed. We will first describe an eample network that will help us focus our thoughts on the particular features that we model, present the characteristics of the model, and then derie the particular mathematical constructs of these characteristics. In section 3, we will present the complete mathematical model, and in Section 4, we will present details of the algorithm to compute the optimal multiserice prices, based on the optimality conditions of the mathematical model. A. The etwork We will use the ring type network shown in Fig., only to highlight the particular features that we are modeling. ote, howeer, that all features we consider in our model are found in any type of network. At each switch there are inputs and outputs of telecommunications traffic. Traffic inputs are from

3 To Appear in IEEE/ACM Transactions on etworking, February many different types of serice that either originate at that switch, or else arrie at the switch from other originating nodes through an incoming bandwidth pipe. Traffic outputs from the switch either go along an outgoing bandwidth pipe, or directly to a user connected to that switch. Without loss of generality with respect to the network characteristics, we assume that buffer space is used at originating switches to collect and smooth data traffic, subect to delay constraints. The data traffic is then transported in bandwidth pipes through the rest of the path to its destination. We allocate aggregate quantities of buffer and bandwidth to connections grouped by serice type, i, origin,, and destination, k. A serice class is therefore denoted by the triple, (i,,k). Each serice type i has traffic characteristics defined by aerage bandwidth r i and peak bandwidth R i. For eample, oice and ideo hae different aerage and peak bandwidth requirements. Traffic Smoothing (Connections Originating at this ode) Data Input from Users Data Output to Users Class Class 2 Class Single Switch View Buffer Space Bandwidth.. Outgoing Bandwidth Incoming Pipe Bandwidth Pipe Incoming Bandwidth Pipe Outgoing Bandwidth Pipe etwork View Fig.. etwork (Ring) Serice Model at an Indiidual Switch. B. Oeriew of the Model The model is deried from the point-of-iew of the serice proider. The serice proider sets prices for the different multiple serices in order to maimize his/her reenue subect to a set of constraints and conditions that are necessary to ensure both quality of serice and flow balance. The problem that the serice proider soles can be summarized as: Maimize Epected Reenue: This is a product of prices charged for each serice and the demand for each serice, taken to be an arrial rate of connection requests, that can be accommodated at that price. Subect to the following constraints: Limited Capacity: The network switches hae limited capacity (bandwidth) Limit the Blocking of Connection Requests: Gien bandwidth capacity constraints, there is a limited number of connections that can be supported in order to guarantee QoS, and some requests may hae to be blocked. We set a limit on the blocking probability for each serice class. ote that the capacity constraint and the limitation on blocking probability will in turn affect the number of connection requests that can be accommodated. Limit the Probability of Packet Loss: Different serices can tolerate different leels of packet loss and still guarantee QoS. We set a limit on the equialent capacity (defined in detail later) allocated to each serice in order to limit the probability of packet loss. Limit the Maimum Delay: Different serices can tolerate different leels of delay and still guarantee QoS. We set a limit on the maimum allowable delay for each serice. We will now describe each component of the model in greater detail using quantities illustrated in Fig.. We will present the complete mathematical formulation of the problem in section II, and derie optimality conditions. C. Deriing Demand The quantity demanded by users, λ, is taken to be the arrial rate of connection requests for serice i, with the connection originating at switch and terminating at switch k. This quantity is determined by a demand function f (p ) where p is the price charged per unit of time the connection is open. Based on current empirical results [][8], in our model we use the demand function below: λ ε a p () where ε is the (constant) elasticity of demand for serice class(i,, k). One important property of () is that marginal reenue with respect to prices will always be negatie (meaning lower prices always increase reenue) proided that ε >. We assume elasticities in ecess of unity for the remainder of the paper. Again, this is suggested by aailable research [][8]. D. Call-Blocking and Epected Reenue We are modeling a network offering connections with guaranteed QoS. Under these conditions, there is a limited number of connections that can be supported, and some requests may hae to be blocked if resources are already resered for other requests. We would like to minimize the blocking of connection requests, and capture this in our optimization model as a constraint. For each serice class, denoted by (i,,k), we model the number of ongoing connections as an M/GI/m/m queue. The capacity in the queuing model, m, refers to the maimum number of connections that we will allow to be admitted to a gien serice class, a ariable we define as. For consistency with this model, we state a number of assumptions: The arrial rate of connection requests, for a serice class, gien by λ (defined aboe) characterizes interarrial times that are independently and identically distributed eponential random ariables. The aerage holding time of a connection, T, is known, and the holding times of indiidual connections can occur according to any general distribution, but are independently and identically distributed for each connection. The traffic intensity of a class, ρ, is the product of arrial rate and aerage holding time, i.e., ρ λ T. The properties of the M/GI/m/m queuing model are well known [4]. The probability of blocking a request within any serice class, BP, is gien by the Erlang B formula:

4 BP P ( C ) n ρ n 0 n! ρ (2)! If the number of open connections, C, is equal to the maimum number permitted for each class,, when a user s request for serice is receied, the connection will be blocked and lost to the system. Otherwise (i.e. C, < ) the request will be admitted. The epected number of busy connections is gien by: E C ρ BP (3) [ ] ( ) We define the epected rate of reenue generation attributable to a serice class, REV, as the price charged per unit time for a single connection, p, multiplied by the epected number of ongoing connections, E[C ]: p E C REV [ ] (4) E. Probability of Loss and Equialent Capacity In the preious section, we modeled the connection leel traffic so we could quantify blocking probability for our model. ow we model the packet leel traffic for the same set of connections, so we can quantify one of the QoS parameters, namely probability of lost packets. This will also allow us to relate the serice classes to the necessary resource assignments, buffer space and bandwidth, to satisfy the QoS. At the switch, packets from all connections of the same class (i,,k) share a FIFO queue of size B. Up to connections, with statistically identical data traffic, may share the allocated buffer at any time. Packet loss, PLOSS, occurs when there is buffer oerflow. To ensure that the guaranteed loss probability is satisfied for each serice class we require that the minimum bandwidth allocation per offered connection be at least equal to the equialent capacity. Equialent capacity, c, for a single source (connection) is defined as, the serice rate of the queue that corresponds to a gien (packet) loss, PLOSS [9][28]. If bandwidth in ecess of the equialent capacity is assigned to a connection, the obsered packet loss is less than PLOSS. There are two main effects that determine the equialent capacity of a single source. Effectie bandwidth refers to the fact that smoothing of data traffic in the buffer reduces the bandwidth required to achiee a prescribed loss rate of packets. Multipleing gains refers to the gain in bandwidth required due to the statistical effect of miing the data traffic of independent connections. We use the term equialent capacity when referring to the combination of these two effects on bandwidth assignment (per connection) to a number of connections with guaranteed QoS. We now state some general properties of equialent capacity. In our model, all connections of a serice class (i,,k) share a single FIFO buffer. Consider a connection of a gien serice type, i, which is distinct from serice class where we include origin- destination information as well. The user is either sending data at a peak transmission rate, R i, (i.e. the on state) or is sending no traffic at all (i.e. the off state). To Appear in IEEE/ACM Transactions on etworking, February The aerage rate of data transmission for the indiidual connection is gien by r i. The on and off periods are assumed to be eponentially distributed, with the aerage length of an on periods gien by b i. QoS requirements for this type of data traffic model hae been well studied, e.g. [9]. The traffic from all connections of the same type, i, is statistically identical, i.e., all connections classified as the same type hae the same parameters r i, R i, and b i. Clearly, the equialent capacity must be greater than or equal to the mean rate of data traffic, r i, and less than or equal to the peak rate of data traffic, R i. r c, B, PLOSS R (5) i ( ) i As figure 2 shows equialent capacity is decreasing in the amount of allocated buffer space, B. c(, B, PLOSS ) 0 (6) B As the number of connections for which we allocate resources,, increases, equialent capacity will decrease reflecting multipleing gains. c(, B, PLOSS ) 0 (7) Finally, as we allow greater loss probabilities, PLOSS, the equialent capacity will decrease. c(, B, PLOSS ) 0 (8) PLOSS An in-depth analysis of equialent capacity is outside the focus of this paper. Detailed discussions of buffering and multipleing gains are gien in [2][9][28][32] and [34]. Howeer, it should be noted that our approach remains unchanged for any deriation of equialent capacity, which satisfies the general properties for equialent capacity, gien in (5) (8) aboe. The inequalities admit special cases such as constant bit rate traffic, with no smoothing or multipleing gains. As we shall see in the net section, the data traffic model and equialent capacity epressions in (5) (8) are sufficient for formulating packet loss in our reenuemaimizing model. It will become apparent later that we wish to sole buffer assignment and bandwidth assignment simultaneously. Therefore, we now illustrate some etensions of the equialent capacity properties when maimum buffer delay is held constant, i.e. the buffer is sized according to a constant maimum delay gien by d B/. Fig. 2 shows the general relationship between equialent capacity (aerage bandwidth per connection) and number of connections when maimum delay in the buffer is held fied. This relationship is deried based on (5) (8) aboe (the serice class indices, (i,,k) are omitted for clarity.) ote that equialent capacity, as illustrated in Fig. 2 below, gies the bandwidth required to meet two QoS criteria, namely loss probability, PLOSS. and maimum delay, d.

5 To Appear in IEEE/ACM Transactions on etworking, February Bandwidth per Connection CMA X B dcma CMA X X CMA B dcma CMA X X X ote CMA X : CMA c( CMA X, B, PLOSS) c ( CMA X, B, PLOSS ) Buffer Space (B) < X Bandwidth per Connection R r ( ) c ( i, i d, i, i PLOSS i ) umber of Connections () Fig. 2. Illustration of Aerage Bandwidth Allocated per Connection, Using Mapping from to. The aerage bandwidth allocated per connection, as a result of the mapping from B to gien by (5) (8) is illustrated by the downward sloping cures in the left panel aboe. An increase in the number of connections sered, from to results in the downward shift the the aerage bandwidth assignment per connection. The straight lines represent the relationship bewteen buffer, B, and bandwidth, for fied maimum delay, i.e. db. Since the figure is drawn in a space representing aerage bandwidth per connection, /, the shift outward in the slope of the lines results from the same increase in the number of connections sered, from to. Collecting the solutions to such a system of two equations in two unknowns yields the downward sloping cure in the right panel of Fig. 2. The figure aboe is conceptual. In our numerical eample, we will use the equialent capacity results from [9] to calculate the equialent capacity per connection summarized below: µ + α σ ( ) c, B, PLOSS min cˆ, (9) where, ( αr B ) 2 αri B + i + 4αB ri cˆ (0) 2α r ( ) i α ln PLOSS bi Ri () µ r i (2) σ ri ( Ri ri ) (3) ( ) ln( ) α 2 ln PLOSS 2π (4) The equialent capacity per connection, (9), is calculated as the minimum of two distinct approimations. The effectie bandwidth is approimated by (0). The second term in the minimum epression in (9) reflects multipleing gains. This approimation is based on the stationary bit rate. To calculate this epression, we need the additional epression (), as well as the mean of the aggregate bit rate, (2), the standard deiation of the aggregate bit rate, (3), and an approimate inersion of the normal distribution, (4). The calculations aboe separate equialent capacity into regions dominated by either smoothing effects in the buffer or multipleing gains. These epressions are one eample of suggested computational methods for equialent capacity that satisfy (5) (8). F. Delay The second QoS parameter that we consider is the maimum allowable delay, d for each serice class. We ignore transmission delay and consider only delay in the buffer. Therefore we can set bounds for the potential delay a packet may eperience quite easily: B d (5) where B is the buffer space and is the bandwidth allocated to serice (i,, k). The maimum delay any packet may eperience, gien by (5), is simply the size of the allocated buffer space diided by the allocated bandwidth. The buffer is sered on a first-in first-out (FIFO) basis. A. The Model III. THE OPTIMIZATIO MODEL Gien the deriations aboe, we are now in a position to present the complete mathematical formulation of the reenue maimization model that would yield optimal prices and resource allocations. We wish to sole for the price, p, for each serice class denoted by (i,,k), as well as the olume of serice offered,. The olume of serice offered refers to how many connections we can sere simultaneously, gien the bandwidth,, resered all along the path between and k, and buffer space, B, resered at the origin switch. We assume that bandwidth is limited but buffer space is not. The network serice is assumed to be offered using a connection admission policy, so that the QoS requirements for probability of loss, PLOSS, and delay, d, are satisfied within certain limits. We include the blocking probability, BP,, resulting from the use of a connection admission policy to ensure that we hae sufficient resources set aside for serice class (i,,k), such that a satisfactory proportion of connection requests can be admitted when they are requested. The total epected reenue from all serice connections in the network is gien by ( i k }. Because the epected number of connections arrials C is a function of the arrial rate, the blocking probability, and the aerage holding time of a connection, we can replace the obectie p E{ C ( p f {λ, BP, T }) function with. i k Thus the reenue optimization model that will yield optimal prices and resource allocation is gien by (Pet) Ma p, B,, i k subect to, ( p f, BP, T }) {λ (6) )

6 To Appear in IEEE/ACM Transactions on etworking, February BP c ( ( p ), T, ) BP λ, (7) i,, k ( B, PLOSS ) B,, (8) i,, k d, i,, k (9) IVP i k IVP, (20), if VP, (2) 0, otherwise i,, k, λ 0, p 0, 0, B 0 (22) 0, integer (23) where, REV rate of reenue generation associated with serice class (i,,k) c (.) equialent capacity of a single connection VP {set of all in the path for class i originating at and terminating at k, } BP maimum blocking probability for a connection from serice class (i,,k) PLOSS maimum packet loss probability for a for a connection from serice class (i,,k) d maimum delay for a for a connection from serice class (i,,k) the capacity (bandwidth) at switch In this formulation, the obectie function, (6), seeks to maimize the aerage rate of reenue generation from ongoing connections, which is gien by price multiplied by the epected number of connections. The constraints restrict the performance measures and resource assignments to be within bounds set outside the problem as network policy. Budgets on each quantity are denoted by a bar oerhead. Constraint (7) restricts the call-blocking probability for eery class below some prescribed limit. Constraint (8) ensures the probability of loss for each serice class, PLOSS, is satisfied. The buffer delay for each serice class is constrained to be less than or equal to the limit gien by the QoS guarantee in constraint (9). We hae a bandwidth capacity constraint at each switch, (20), but we assume there is no capacity on allocated buffer space, and there are no link capacities. The capacity at each switch must accommodate all traffic originating at the switch as well as traffic originating elsewhere but routed through the switch. We assume the routes are known and fied. There is an indicator function, (2), for eery class (i,,k), which indicates if any switch is included in the path. There are a number of non-negatiity constraints, gien by (22). Finally there is an integrality constraint on the maimum number admitted to each class,, gien by (23), since connections can only be admitted in discrete quantities. We wish to call attention to one limitation of our formulation of the capacity constraints, which may oerassign bandwidth along the paths followed. The effectie bandwidth of the aggregate traffic for the serice class (i,,k) may be reduced after smoothing in the buffer at the originating switch. Our formulation assigns bandwidth all along the path based on the effectie bandwidth at the originating switch and does not take into account the statistical properties of the output traffic at the originating switch. The simplicity of the formulation outweighs our concern regarding ecess bandwidth assignments. B. Optimality Properties In general, the problem gien in (Pet) is a nonlinear, noncone, mied integer problem. Howeer, there are a number of necessary conditions for an optimal solution, which we will use to search for a solution. First we will discuss necessary conditions related to constraints (7) (9) in problem (Pet). These constraints are relatiely simple as they apply to each serice class independently of all other serice classes. Based on this first set of necessary optimal conditions, we will then state a pair of optimality conditions reflecting the marginal alues of each serice class (i,,k) in the solution. ) Resource Allocation to Indiidual Serice Classes There are a number of conclusions we can immediately draw about optimal solutions to the problem (Pet). First we consider the resources that must be assigned to each indiidual serice class. Theorem: Gien downward-sloping demand cures and plentiful buffer space at each originating switch, if an optimal solution to (Pet) does not coincide with the condition that marginal reenue equal to zero, i.e. the partial deriaties of () with respect to prices are all less than zero at optimality, then the optimal solution to (Pet) must satisfy the following properties: ( ( p ), T, ) BP BP λ, (24) k i,, k (, B, PLOSS ) c, (25) i,, k ( + ) c ( +, B, PLOSS k ) + i k i k k k k > ( ) c, B, PLOSS, (26) B d, i,, k (27) Proof: See Appendi for proof.

7 To Appear in IEEE/ACM Transactions on etworking, February The call-blocking probability for all serice classes is set to its greatest permitted alue for all serice classes in (24). Incidentally, this is the only condition, which pertains directly to prices. The bandwidth assigned to each serice class, for the maimum number of connections admitted, will be equal to the corresponding equialent capacity, as stated in (25). The third condition, (26), states that because connections hae to be integer, allocation of the equialent capacity for any additional connections would iolate feasibility. This means that no further increases in bandwidth assignments are possible at an optimal (feasible) solution. Buffer space is assigned for all serices originating at each switch, such that packets for each serice may eperience a delay up to the tolerated delay, (27), assuming that buffer space at each switch is plentiful. The optimal resource allocation is to assign the appropriate effectie bandwith to each serice class, (25), and buffer space proportional to the equialent capacity (27). Typically, buffer space is thought of as a parameter in the equialent capacity calculation, while we are treating it as a ariable (along with bandwidth), for which we sole a system of two equations in two unknowns. That is, we sole (25) and (27) for and B : ( ) (, d PLOSS ) : c, (28) Going back to Fig. 2 which shows equialent capacity as a decreasing function of buffer space, and the number of connections sered, the optimality property, (27), is illustrated with a linearly increasing function of buffer allocation, where both sides of the epression hae been diided by. As the number of connections sered increases, the slope of the line becomes less steep. Based on this simple graphical analysis, the system of equations, (25) and (27), must hae a unique solution and be decreasing in terms of. ote that if there are no smoothing effects of multipleing gains, as with constant bit rate traffic (R i r i ), then the allocation per connection is simply a constant alue. In all other cases, the total allocation must therefore be increasing but the allocation per connection may be decreasing, depending on the properties of the serice class. This shows economies of scale in the optimal resource allocation. Using another optimality property from Theorem, the call-blocking constraint will be binding by (24). We can sole for the the optimal arrial rate, λ, for a gien, using (): ρ ρ ρ ( ) ρ : BP (29)! n! ρ λ ( ) (30) T u + ( ) n 0 The arrial rate, λ, is determined by the limit on callblocking. We must first calculate the maimum traffic intensity for, gien in (29), and then calculate the n arrial rate, using (30). Because the resource allocation tables are independent of demand we hae not yet related the arrial rate to a price. The bid tables, presented in the net section, will relate the price required, p, to produce the desired arrial rate, λ, as well as the marginal aluation at the olume of serice proided,. 2) Resource Allocation Trade-Offs Between Serice Classes The optimality properties, from sub-section ) aboe, reduce the problem to choosing the optimal alues for. For the time being, consider a linear relaation to (Pet), eliminating the integrality constraints on. For solutions satisfying the optimality properties gien by (24) (27), the Karush-Kuhn-Tucker necessary conditions are triially satisfied, with the following eception, which results from differentiating the set of constraints (20): REV IVP i,, k, (3) Recall that IVP in (3) was defined as an indicator parameter in the problem, (Pet). For each serice class (i,,k), we diide both sides of (3) by /, and simplify the necessary optimality conditions: u IVP, i,, k (32) Economic interpretation of (32) is as follows: Any serice class must yield a marginal return per unit of bandwidth equal to the sum of the marginal alues for all switches ( for switch ) along the route for the gien class (i,,k). Incorporating the integrality requirement in, we define marginal alues per unit of bandwidth for increasing or reducing the number of connections in the solution, u + and u - respectiely: ( ) u+ REV ( + ) REV ( ( + ) ( ) ) (33) u ( ) REV ( ) REV ( ( ) ( ) ) (34) The marginal aluations, u - from (33) or u + from (34), are simply the changes in epected reenue diided by the change in bandwidth allocation, for an increase or decrease of one in. ote that by definition, u + ( ) equals u - ( + ). Discrete approimations of the continuous necessary optimalty conditions (32) are: IVP u i,, k ( ), (35) In stating (35), we assume that marginal reenue is

8 To Appear in IEEE/ACM Transactions on etworking, February decreasing in, i.e. u - > u +. By (35) it is not profitable to change the bandwidth allocated to any serice class. The marginal alue from an increased allocation is less than the sum of marginal alues of bandwidth at switches along the path. IV. ETWORK AUCTIO ALGORITHM The search procedure for the global network solution seeks to identify the marginal aluations of bandwidth at all switches,, that will maimize epected reenue. Based on the optimality conditions (24) to (26), it is clear that, the search will hae to test and adust marginal alues at each switch until we find a solution that allocates bandwidth fully at each switch in order to satisfy the optimality conditions. A. Storage of Problem Data in Bid Tables We can eploit the first set of necessary optimality properties aboe, (24) (27) to simplify the search space for the the problem (Pet). These properties dictate that optimal allocations of can be calculated from (28) based on the alues of. There is a unique arrial rate of connection requests, λ, associated with, according to the call-blocking property, (29). In turn prices, p, are related to alues of through the arrial rate, λ, gien by the demand functions, (). This gies us a reduced space of problem data, where all other problem ariables are calculated as functions of, which are integer-laued ariables. Furthermore, we hae defined marginal alues of serice classes, u - and u +, for any alue of, in (33) and (34). We can calculate all the ariables referred to aboe off-line and summarize the search space of the problem in tables indeed by the integer-alued ariables, (Table ). We call Table I a Bid Table for the following reason: The marginal alues u + (or u - ) represent the maimum amount a rational agent would bid (or accept) per unit of bandwidth to add (or remoe) a connection of serice class (i,,k). ote that the bid table contains much less information than all the feasible alues of,, B and p. thus making it much faster to sole problem (Pet). TABLE I DEFIITIO OF A BID TABLE eq. (28) B eq. (27) λ eq. (30) p eq. () + u eq. (33) u eq. (34) + () B () λ () p λ () u () u () ( ) + 2 ( 2) B ( 2) λ ( 2) p λ ( 2) u ( 2) u ( 2) ( ) M M M M M M M B. Retrieal of Problem Data from Bid Tables For any gien set of marginal bandwidth aluations,, we look up the bid and the associated number of connections and resource allocations. Strictly speaking, we select for a gien according to the following rule: (, 2,, ) K + ma : u < IVP < u (36) i We search from the bottom of the bid tables and take the first (and largest) alue of for which u + < < u -. For relatiely small alues of the u - /u + alues may be increasing due to large multipleing gains. Since we are interested in reenue maimization, (36) selects the largest alue of, which satisfies the optimality property, (35). The simplest way to find this alue is by looking up the bid table data starting from the bottom of the table. We will now present a method called the auction algorithm for ensuring that the we choose satisfies the optimality conditions. C. Bounds on Marginal Value of Bandwidth For conenience in describing the search procedure, we define bounds on the optimal set of marginal aluations of bandwidth for all switches: {,, K, } {, 2, K } < {, 2,, 2 K < } (37) where, optimal marginal aluation at switch a lower bound on the alue of at switch an upper bound on the alue of at switch Clearly, using the bid table and (37), we can guarantee a set of marginal alues is either an upper bound or a lower bound on an optimal set of marginal alues: i k i k (, ) IVP >, K (38) 2, (,, K ) IVP < (39) 2, A set of marginal alues that underalues the bandwidth at all switches results in an oer-assignment of bandwidth at all switches and be infeasible. Thus, we can use such a set of marginal alues as a lower bound on the set of optimal marginal alues. Likewise a set of marginal alues which oeralues the bandwidth at all switches will be feasible and can be used as an upper bound on an optimal set of marginal alues. ote that while we offer bounds on the optimal alue of in (37), we claim only to be seeking bounds for a local optimal solution to the problem (Pet), which is non-cone and contains integer-alued ariables. D. Search Procedure Our search method begins with arbitrary bounds on the optimal marginal aluation of bandwidth at each switch, which satisfy (38) and (39). Each iteration, we decrease the Euclidean distance between the two sets of bound and change the direction of the line segment between the two sets of bounds in the marginal alue space. When the upper and lower bounds are ery near to each other, and we hae a feasible solution that fully assigns capacity at all switches, we terminate the search and obtain a near optimal solution, where

9 To Appear in IEEE/ACM Transactions on etworking, February capacity is fully assigned and reenue is ery close to optimal. The infeasible solution gien by the lower bounds on marginal alue of bandwidth at each switch will also proide a bound on the distance from a local optimal solution. The search algorithm is as follows: Step Initialization: For initialization, we simply require an initial set of bounds on the optimal marginal alues as defined in (38) and (39). Step 2 Identifying Under-Valued (and Oer-Valued) Bandwidth: We wish to identify the best feasible and worst infeasible solutions that lie along the line segment between the current upper and lower bounds on marginal alue. The bid table data is calculated according to optimality properties on resource assignments, and the constant elasticity of demand model implies higher reenue for lower prices and more connections proided. Therefore, the best feasible solution along the line segment between the upper and lower bounds on marginal alue is where the largest number of connections are proided by assigning the maimum possible capacity. This in turn is obtained at the lowest feasible marginal alues. Similarly, the worst infeasible solution along the line segment is found with the highest marginal alues along that line, for which the capacity assignment is infeasible at eery switch. We sole the following line search problems to obtain these two cases: (P-feasible) Ma α (40) subect to, i k (P-infeasible) Ma β (42) subect to, i k where, m ( αm, αm, K αm ) IVP 2 2, < (4) ( + βm, + βm, K + βm ) IVP 2 2, > (43) (44) 0 < α, β < (45) The line segment between the current set of lower bounds and upper bounds is gien by (45). The maimum reenue feasible solution along this line is gien by the set of marginal alues - α m. The minimum reenue infeasible solution is gien by + β m. We will use these solutions in the net step of the algorithm to determine which bounds should be changed before the net iteration of the algorithm. Step 3 Adusting Lower and Upper Bounds on Marginal Value: Along the line segment between the upper and lower bounds, we wish to identify the switch for which the bandwidth which is most under-alued. That is, the switch(es) for which marginal alue of bandwidth is too low at the solution α to the reenue maimization problem (P-feasible) aboe, yielding the most assigned bandwidth. Similarly, we wish to select the switch (or switches) for which bandwidth is most oer-alued, or the switch for which the lower bound is too high at the solution β to the problem (P-infeasible) aboe, yielding the least assigned bandwidth. These alues are defined mathematically below: under alued oer alued argmin i k α m, K IVP α m : + β m (49),, (46) + β m, K argma IVP (47) i k + β m The determination of which switches are oer or underalued in terms of their marginal alues of bandwidth is the same as simply choosing the switch with the highest assigned bandwidth from the solution to (P-feasible) and that with the lowest assigned bandwidth to (P-infeasible). For eample, if one switch is 00% assigned at the solution to (P-feasible) while all others are less than 50% assigned, the marginal alue of bandwidth at the fully assigned switch is relatiely too low for an optimal solution, since capacity should be fully assigned at all switches. ote that ties are permitted so that there may be more than one under-alued or oer-alued switch. To effectiely raise the marginal alue of bandwidth when soling (P-feasible), we will increase the lower bound on marginal alue at that switch. When we then repeat the line search to sole (P-feasible) in the net iteration, a higher marginal alue will result at that switch relatie to the other switches. Similarly, we will decrease the upper bound on marginal alue for the switch that has the lowest infeasible assignment at the optimal solution to (P-infeasible), e.g. lower the upper bound at a switch with 0% assignment when all others are ~50% assigned. This effectiely lowers the marginal alue calculated in the nest iteration where we sole (P-infeasible) again. oer alued under alued : α m (48) oer alued under alued oer alued under alued We raise the lower bound on marginal alue for the underalued switch to the alue gien by the solution to (Pinfeasible), as gien in (49). Similarly, we decrease the upper bound on marginal alue for the oer-alued switch by taking the alue in the solution to (P-feasible), as gien by (48). ote the use of an assignment operator, : in (49) and (48). The alue to the left of : is the new alue being calculated, while the alue to the right of : is based on the preious quantities.

10 To Appear in IEEE/ACM Transactions on etworking, February Step 4 Termination Test: When the line search between the bounds yields a solution to (P-feasible) where capacity is fully assigned at all switches, we terminate the algorithm, and assign the alues found in the line search problem (P-feasible) as the solution. If, i k then, m α, K, αα m IVP (50) α m. (5) Terminate the search. Else, Go to Step 2. There are a couple of things to note about this iteratie algorithm. First, the algorithm is really a bisection search as described in Step 3, where the gap between lower and upper bounds reduces at eery step. It is well known that bisection searches are guaranteed to conerge. We also show this numerically (See Figure 4 later). et, because our search space is contained in the bid tables we hae described earlier, which are indeed by integer ariables,, our search space is lumpy. As such, fully assigned means an arbitrarily chosen leel of capcity utilization such as 99% assigned capacity at eery switch. This is why we use the notation approimately equal,, rather than requiring strict equality,. The final line search problem (P-infeasible) yields an upper bound on the obectie alue. That is for the local optimum gien by (5), and the solution to (P-infeasible) we can calculate the epected reenues and state the duality gap for the local near-optimal solution is gien as follows: REV ( ) + β m, K, + β m > 0 (52) ( ) i k REV + α m, K, + α m V. EXAMPLE PROBLEM We will now present a simple real world problem with twoserice classes and show how to set optimal prices and resource allocations using the model presented in section 3 along with the search method presented in section IV. A. etwork Structure and Serice Classes Consider a single network offering oice and ideo connection to users ia a set of fie switches. We consider a bi-directional ring network (Fig. 3). The oice connections can be either local, i.e., routed through a single switch, or long-distance to anywhere in the network, i.e. routed from any origin to any destination switch. The ideo connections originate from a single switch (labeled in the figure), where a ideo serer is aailable and may be routed to any switch in the network, including the originating node. The connections are all routed using the minimum hop routing. Switch capacities are chosen as follows. Switch 0 is included in the path for eery ideo connection and has 5000 Mbps capacity, or fie times as much capacity as switches 2 and 3, which are each included in one fifth of the paths for Mbps Video Serer Switch Mbps Switch 0 Switch Mbps 2000 Switch Mbps Switch Mbps Fig. 3. Two serice network with a single ideo serer Serice Class TABLE II SERVICE CLASS DEFIITIO Mean. Rate (Mbps) r i (a) Data Traffic Parameters Peak. Rate (Mbps) R i Aerage Burst (s) Voice Video Serice Class Packet Loss Probability PLOSS i (b) Other Parameters Blocking Probability BP i Delay (s) d i b i Aerage Holding Time (s) Voice Video ideo connections. Similarly, switches and 4 hae two times the capacity of 2 and 3. The network serices offered are oice and ideo. The serice class definitions are gien in Table II, which also proides traffic data Voice serice is a reflection of traditional oice serice, which does not hae a high peak bandwidth, and has relatiely short periods of bursts. In terms of QoS, relatiely high loss rates of packetized oice may be acceptable, but large delays cannot be tolerated,. Voice connections are typically of short duration (e.g. T oice 3.0 minutes), which results in a lower aerage number of connections in use for any arrial rate of requests, as defined by (3). Video is intended to reflect bursty sources of data traffic, where the bursts, such as action scenes can be a high peak and continue for prolonged periods. The peak and mean data rates are defined conseratiely, based on MPEG- trace data in [29]. The length of the idle/busy period is chosen T i

11 To Appear in IEEE/ACM Transactions on etworking, February 2003 based on a suggestion in [34] that ideo sessions see busy periods in the order of 0 seconds. For users with buffer space at the client end, a modest delay in the transmission of packets is acceptable. Some packet loss can be concealed, but a higher degree of reliability is required than for oice. We assume constant elasticity of demand for both serices, with the elasticity alues taken from [8] (Table III). ote that, the elasticity of ideo is higher than that of oice. This means that the reenue, associated with ideo connections, increases more rapidly as the price is lowered than for oice connections. We hae chosen the scaling factor in the functions to be and 2 respectiely. This means that at the price (or a normalized price) the arrial rate of requests Serice Class TABLE III DEMAD FOR SERVICES Demand Voice (,k) λ oice,k.0p -.05 Video ( ) λ ideo,k 2.0p Video ( ) 0 for ideo is twice as high, which reflects the higher alue of a connection with ideo than that with simply oice. B. Illustration of Solution Algorithm Fig. 4 and 5 proide details of the search steps when we implemented the algorithm for the Eample Problem aboe. As we can obsere, the search starts with lower and upper bound alues at ectors 0 and respectiely, and through a number of bi-section steps, conerges to the best (i.e. near optimal) solutions at iteration 8. All other ariables also conerge. To clarify eactly how the tight bounds are generated by the search method, we eplain the step through an iteration in the Fig. 4 eample, before continuing with the interpretation of the solution to the same eample. A Single Iteration of the Search Algorithm: At iteration 0, Fig. 4 shows the starting upper and lower bounds on node marginal alues. Recall from (32) that the marginal reenue per unit of bandwidth (u + ) for a gien serice class must offset the sum of marginal alue(s) of bandwidth () at eery along the path assigned to the serice class. At each estimate of node marginal alue, bandwidth assignments etc. will be obtained from the bid table. Thus, at the upper bounds on node marginal alues, resource assignments will be ery small and feasible. Conersely, at the lower bounds there will be infeasible oer-assignment of resources. This is shown in Figure 5. At the initial upper bounds (,,,,), bandwidth assignments at each node are (26.42%, 2.66%, 5.32%, 5.32%, 2.66%) of aailable capacity. At the lower bounds on node marginal alues (0, 0, 0, 0, 0), the assignments are (899.32%, %, 37.74%, 37.74%, %) of aailable capacity. (These are not shown because they are outside the scale of Fig. 5). Marginal Value of Bandwidth Se arch Illustration Bounds on M arginal Values Iteration 0 After 0 After 2 After 2 3 After 3 4 After 4 Fig. 4. Search Algorithm Steps: Conergence of Marginal Values Percentage Assignment of Bandwidth 300% 250% 200% 50% 00% 50% 0% Search Illustration B ounds on Capacity Assignments Iteration 0 After 0 After 2 After 2 3 After 3 4 After 4 Fig. 5. Search Algorithm Steps: Conergence of Capacity Assignments At iteration, a bisection along the line segment oining (0, 0, 0, 0, 0) and (,,,, ) is obtained. If feasibility is not important, the maimum distance found along that line segment yields node marginal alues to be (0.40, 0.40, 0.40, 0.40, 0.40); see Figure 4. In this case, the bandwidth assignments are (285.73%, 50.39%, 00.0%, 00.0%, 50.39%) of capacity (now shown in Figure 5). When feasibility has to be maintained, the bisection search yields node marginal alues of (0.64, 0.64, 0.64, 0.64, 0.64), for which the bandwidth assignments are (99.98%, 43.04%, 8.0%, 8.0%, 43.04%); see Figure 5. Gien the broad range of capacity assignments (from 8.0% to % of node capacity in the feasible case and from 00.0% to % of node capacity in the infeasible case), we hae an intuition that the bounds on the node marginal alue either under-assigns and oer-assigns bandwidth at certain nodes in a relatie sense. ode 0 with an infeasible assignment of % of capacity in this iteration is most

12 To Appear in IEEE/ACM Transactions on etworking, February oer-assigned, or what we now call, under-alued. Therefore, we raise the marginal alue for node 0 relatie to all the other nodes by resetting the lower bounds from (0, 0, 0, 0, 0) to (0.40, 0, 0, 0, 0). Likewise, the assignment at the tightest feasible solution is most under-assigned or oeralued at nodes 2 and 3, where only 8.0% of aailable capacity is included in the current feasible solution. Therefore, we lower the relatie marginal alue of nodes 2 and 3 by resetting the upper marginal alue bounds from (,,,, ) to (,, 0.64, 0.64, ). The new bounds are passed to the algorithm for the net iteration and define the bounds at the beginning of iteration 2 in Fig. 4. The bisection search along the line segment between bounds on marginal alues proceeds as aboe until conergence. C. Solution Interpretation The solution is found when the algorithm conerges to a set of marginal alues on indiidual node bandwidth, which TABLE IV OPTIMAL PRICE PER UIT TIME FOR A COECTIO OF SERVICE (i,,k). Serice Type Origin Destination (k) (i) () Voice Video occurs after 8 iterations in Fig. 4. The final marginal alues of the nodes selected for the near optimal solution (i.e. the final solution to (P-feasible) as discussed in in Step 4 of Section IV are (0.97, 0.4, 0.20, 0.20, 0.4). The epected reenue at this solution is The duality gap (calculated by comparing the solution to (P-feasible) with the solution to (Pinfeasible)) is roughly 0.24%; i.e. the best feasible solution we found was within 0.24% of the optimal solution. The bandwidth assignments at eery switch are in ecess of 99%, the criteria we used for termination in S4; i.e. the optimality conditions are, more or less, satisfied. Gien the marginal alues aboe, the optimal prices are simply looked up in the bid tables and are gien in Table IV. The prices reflect relatie scarcity of bandwidth at switch, due to the location of the ideo serer. Relatie to switch, local (same origin and destination) oice connections are priced inepensiely elsewhere in the network. The longdistance connections must yield much higher marginal alues than local connections to be profitable, since the long distance connections must outweigh the marginal alue of local bandwidth all along the path of the connection. High marginal alues correspond to higher prices and relatiely smaller allocations. The ideo connections at switch are the least epensie. Video connections at switch use only resources at switch, while all other ideo connections must use this resource plus more resources at other switches. Consequently the olume of ideo connections at switch must also be the highest of anywhere in the network. Howeer, the cheaper ideo serice comes at the price of epensie oice serice through this switch, because the oice connections oer this switch must use resources with a ery high marginal alue relatie to the other switches. D. Comparison to Flat Rate Pricing Approaches To ealuate the usefulness of our approach, we compared optimal pricing with two flat rate pricing, per-hop and perconnection pricing, approaches. In the current US oice market, pricing per connection regardless of origin and destination is ery popular. Pricing-per-hop is another flat-rate pricing scheme that has been proposed, and mimics the use of prices proportional to link use. For both of these pricing approaches, we need only choose a price for each serice type in the network, a total of prices. Because we hae only two serice types, we assume a ratio between the oice price and ideo price and then simply sole for the smallest oice price that achiees feasibility at all switches. We will use the bid tables, with the resource allocations satisfying optimality properties and find the smallest feasible flat rate price for oice, subect to the assumed ratio between oice and ideo prices. Results are summarized in Figs. 6 and 7. The optimal pricing mechanism proided greater reenue than either flat-rate pricing approach. The saings were upward of 20% in all cases. In general, pricing per hop produces higher reenue than pricing per connection in the reasonable ranges for relatie ideo and oice prices. In this range, the epected reenue is relatiely insensitie to the ratio between the prices for the two serices. The pricing per connection shows the highest profit when oice and ideo connections are charged at the same rate. The prices essentially fill the network bandwidth with ideo connections, which are highly elastic and generate significant reenue as Percent Improement with Optimal Pricing 40.00% 35.00% 30.00% 25.00% 20.00% 5.00% 0.00% Decrease in Epected Reenue with Pricing Per Hop ersus Optimal Pricing 5.00% 0.00% Ratio of Prices Per Hop (Video/Voice) Fig. 6. Comparison of per hop pricing with optimal pricing.

13 To Appear in IEEE/ACM Transactions on etworking, February Percent Improement with Optimal Pricing Decrease in Epected Reenue with Flat Rate Pricing ersus Optimal Pricing 40.00% 35.00% 30.00% 25.00% 20.00% 5.00% 0.00% 5.00% 0.00% Ratio of Prices Per Connection (Video/Voice) Fig. 7. Comparison of per connection pricing with optimal pricing. prices are lowered. Howeer, users are unlikely to accept a oice rate as high as would be necessary for a ideo connection. For this reason, the solutions with larger ratios between the ideo price and oice price are probably more practical. These solutions earn significantly less reenue. VI. COCLUDIG REMARKS This paper presents a mathematical programming model for optimal pricing, and bandwidth and buffer allocation for multiple serices with QoS guarantees in a connectionoriented network. We also present a noel solution methodology. We show, using numerical eperiments, that falt-rate pricing (whether by connection or hop) is inferior to the multi-serice pricing obtained from our model. Our model proides a ery powerful mechanism for pricing multiple serices in communications networks. In order to implement our pricing scheme in a practical real world setting, the first decision to be made is whether one uses a centralized architecture or a decentralized one. In either case, a processor has to use demand (i.e. connection request) information to make proections of future demand for the different types of serices. In a centralized architecture, the master switch will directly get information about capacity aailabilities at each switch. Using this and the demand proections, the master switch will calculate node marginal alues for each switch, and will determine the prices to be set and the resources to be allocated for each serice class, updating it at reasonable time interals. The model presented in this paper is particularly useful in this setting. In a decentralized architecture, each switch will also hae to calculate the prices and resources needed for each class of serice. In order to hae globally optimal solutions, each switch will hae to communicate information pertaining to itself to the rest of the switches. Depending on whether the communication is sent only to neighboring switches, or to all of the switches, the actual calculations at any point in time might differ. The model we hae presented in this paper will still work, if all of the decentralized switches use it for pricing and resource allocation, but obtain perfect instantaneous information from the other switches on node marginal alues and demand for bandwidth. Howeer, if there is partial information and/or a significant time-lag for the communications protocol to yield up-to-date perfect information, a simultaneous implementation of the model by all switches might not yield a globally optimal solution. We are currently working on multiple serice pricing mechanisms for decentralized networks. We are also working on obtaining good demand proections, which is a further limitation of this paper that assumes that all parameters of the stochastic arrial process is known. The beginning of such work was reported in Keon s dissertation [5]. APPEDIX: OPTIMALITY THEOREM We argue for each of the optimality properties in the theorem indiidually: i. Assume there is an optimal solution with prices and resource allocations such that the call-blocking constraint is non-binding, for at least one serice class: ( ( p ), T, ) BP BP λ < (53) It follows that for this class the network operator could allow a higher rate of connection requests and still satisfy the call-blocking constraint: λ p > λ p : BP λ p, T, BP ( ) ( ) ( ( ) ) (54) Demand is downward sloping, and marginal reenue with respect to prices is less than zero by assumption, so that the price corresponding to the higher rate of requests must be lower and reenue must be higher: dλ dp ( p ) < 0, p < p ( ( p ), T, ) BP (55) BP λ (56) Therefore, for (53) an optimal solution cannot eist. The optimal solution must be such that call-blocking is binding: ( ( p ), T, ) BP BP λ (24) Assume there eists an optimal solution such that the allocation of bandwidth to a particular class is greater than the equialent capacity for that class: c (, B, PLOSS ) < (57) There must eist a feasible allocation of less bandwidth, according to constraint (8) in (Pet) (the smallest feasible assignment is equal to equialent capacity at the maimum permitted loss probability) for the particular class such that the QoS is still satisfied: B W < : c ( B, PLOSS ), (58) The reassignment gien by (58), where equialent capacity, c i, is minimized by selecting the loss probability equal to its

14 To Appear in IEEE/ACM Transactions on etworking, February maimum permitted alue, may make it possible to assign bandwidth for an additional connection, lower the price of such a class and increase reenue. Een if admitting more connections is not possible due to the reassignment in (58), reenue cannot decrease from the reassignment. Therefore, the assumption that bandwidth is ecess of effectie bandwidth, (57), makes no sense and the optimal solution is inalid. We must hae assigned bandwidth equal to equialent capacity: c (, B, PLOSS ) (25) ii. Assume that a solution is optimal and possesses the following property for at least one local serice class, e.g. (,,): ( + ) c( +, B, PLOSS ) ( ) c B PLOSS (59) +,, i k i For the particular serice class (,,), we can allocate resources for an additional connection, and tolerate a higher arrial rate of requests, at a lower price, while satisfying all constraints: λ p > λ p p < p all constraints satisfied(60) ( ) ( ) :, Similarly to aboe, a lower price results in increased reenue. Therefore (59) is false and capacity must be fully allocated: ( k + ) c k ( k +, Bk, PLOSS k ) > + ( ) c, B, PLOSS i k i, (26) We first assume that there eists an optimal solution that satisfies the following property for a single serice class (i,,k), which contradicts condition (27) in Theorem : B d < (6) For this serice class, as we increase the buffer allocation, B, the equialent capacity c, or minimum bandwidth assignment per connection falls with the increased buffer space: c( B ) < 0 (62) B Therefore, there eists buffer and bandwidth allocations different from those gien in (6), for which the bandwidth assignment is lower: B > B, c B < : B d (63) ( ) If the difference in bandwidth is sufficiently large there may be a reenue increasing solution, by (26) aboe, meaning the current solution cannot be optimal. On the other hand if the bandwidth allocation does not yield an improement in reenue, the reenue cannot be decreased by the reallocation described aboe. Therefore, the optimal solution must satisfy (27): B d, i (27) ACKOWLEDGMET We thank without implicating Roch Guerin, Yannis Korilis and Viay Chandru and referees of this ournal for comments and suggestions REFERECES [] Aldebert, M., M. Ialdi and C. Roucolle, Telecommunications Demand and Pricing Structure: An Econometric Analysis, Proceedings of the 7th International Conference on Telecommunications Systems: Modelling and Analysis, pp , ashille, T, March 8-2, 999. [2] Bertsekas, D. and R. Gallager, Data etworks, Prentice-Hall, Englewood Cliffs, ew Jersey, (987). [3] Bertsekas, D., Auction Algorithms for etwork Flow Problems: A Tutorial Introduction, Computational Organization and Applications, ol., 7-66 (992). [4] Bhat, U.. and Basawa, I.V., Queuing and Related Models, Oford Uniersity Press, ew York, ew York (992). [5] Cocchi, R., S. Shenker, D. Estrin and L. Zhang, Pricing in Computer etworks: Motiation, Formulation, and Eample, IEEE/ACM Transactions on etworking, ol., no. 6, (993). [6] Courbetis, C., V. Siris and G.D. Stamoulis, Integration of Pricing and Flow Control for Aailable Bit Rate Serices in ATM etworks, Globecom, (996). [7] Economides, A.A. and J.A. Silester, Multi-Obectie Routing in Integrated Serices etworks: A Game Theory Approach, in Proceedings of IFOCOM 9, pp (99). [8] Edell, R.J.,.McKeown and P.P. Varaiya, Billing Users and Pricing for TCP, IEEE Journal on Selected Areas in Communications, ol. 3, pp , September 995. [9] Guérin, R., H. Ahmadi and M. aghshineh, Equialent Capacity and its Application to Bandwidth Allocation in High-Speed etworks, IEEE Journal on Selected Areas in Communications, ol. 9, no. 7, (99). [0] Gibbens, R.J. and F.P. Kelly, Resource Pricing and the Eolution of Congestion Control, draft paper (999). Aailable: [] Jiang, H. and S. Jordan, The Role of Price in the Connection Establishment Process, European Transactions on Telecommunications Economics of Telecommunications, oember 9 (994). [2] Jordan, S. and H. Jiang, A Pricing Model for High Speed etworks with Guaranteed Quality of Serice, IEEE InfoCom, pp , March 996. [3] Kelly, F.P., Tariffs and Effectie Bandwidths in Multiserice etworks, Proceedings of ITC 94, pp , 994. [4] Kelly, F.P. A.K. Maullo and D.K.H. Tan, Rate Control in Communication etworks: Shadow Prices, Proportional Fairness and Stability, Journal of the Operational Research Society, ol. 49, (998). Aailable: [5] Keon, eil J., Pricing in Telecommunications etworks Offering Multiple Serices and Quality of Serice Guarantees, Uniersity of Pennsylania, Systems Engineering Department, August [6] Korilis, Y.A. and A.A. Lazar, On the Eistence of Equilibria in oncooperatie Optimal Flow Control, Journal of the ACM, ol. 42, pp , May 995. [7] Korilis, Y.A., T.A. Vararigou and S.R. Ahua, Pricing oncooperatie etworks, submitted to the IEEE/ACM Transactions on etworking, May 997. [8] Lanning, S., D.Mitra, Q. Wang and M. Wright, Optimal Planning for Optical Transport etworks, presented at the Fifth IFORMS Telecommunications Conference, Boca Raton, FL, March 5-8, [9] Low, S.H. and P. Varaiya, A ew Approach to Serice Proisioning in ATM etworks, IEEE/ACM Transactions on etworking, ol., no. 5, (993).

15 To Appear in IEEE/ACM Transactions on etworking, February [20] Low, S.H. and D. E. Lapsley, Optimization Flow Control, I: Basic Algorithm and Conergence, IEEE/ACM Transactions on etworking, Dec [2] Low, S. H., Equilibrium Bandwidth and Buffer Allocations for Elastic Traffics, IEEE/ACM Transactions on etworking, Vol. 8, no. 3, pp , (2000) [22] Mackie-Mason, J.K. and H. Varian, Pricing the Internet, Public Access to the Internet, eds. B. Kahin and J. Keller, Cambridge and London: MIT Press, , 995. [23] Mackie-Mason, J.K. and H.R. Varian, Pricing Congestible etwork Resources, IEEE Journal on Selected Areas in Communications, ol. 3, pp. 4-49, September 995. [24] Mackie-Mason, J.K. and H. Varian, Some Economics of the Internet, in: etworks, Infrastructure and the ew Task for Regulation, eds. W. Sichel and D.L. Aleander, Ann Arbor: Uniersity of Michigan Press, 07-36, (996). [25] Masuda, Y. and S. Whang, Dynamic Pricing for etwork Serice: Equilibrium and Stability, Management Science, ol. 45, no. 6, pp , (999). [26] Mendelson, H. and S. Whang, Optimal Incentie-Compatible Priority Pricing for the M/M/ Queue, Operations Research, ol. 38, no. 5, pp , (990). [27] Paschalidis, I.Ch., and John. Tsitsiklis, Congestion-Dependent Pricing of etwork Serices, IEEE/ACM Transactions on etworking, ol. 8, no. 2, pp 7-84 (2000) [28] Perros, H.G. and K.M. Elsayed, Call Admission Control Schemes: A Reiew, IEEE Communications, ol. 34, no., 82-9 (996). [29] Rose, O., Statistical Properties of MPEG Video Traffic and Their Impact on Traffic Modeling in ATM Systems, Proceedings of the 20th Annual Conference on Local Computer etworks, Minneapolis, M, pp , 995. [30] Shenker, S., Serice Models and Pricing Policies for an Integrated Serices Internet, Public Access to the Internet, eds. B. Kahin and J. Keller, Cambridge and London: MIT Press, (995). [3] Thomas, P., Teneketzis, D. and Mackie-Mason, J.K., A Market-Based Approach to Optimal Resource Allocation in Integrated-Serices Connection-Oriented etworks, submitted for publication, August 999. [32] de Veciana, G., G. Kesidis and J. Walrand, Resource Management in Wide-Area ATM etworks Using Effectie Bandwidths, IEEE Journal on Selected Areas in Communications, ol. 3, no. 6, (995). [33] de Veciana, G. and R. Baldick, Resource Allocation in Multi-Serice etworks ia Pricing: Statistical Multipleing, Computer etworks and ISD Systems, ol. 30, (998). [34] Walrand, J. and P. Varaiya, High-Performance Communication etworks, Morgan Kaufmann Publishers, Inc. San Francisco, California (996). [35] Wang, Q., J.M. Peha, and M.A. Sirbu, The Design of an Optimal Pricing Scheme for ATM Integrated Serices etworks, Special Issue: Internet Economics, Journal of Electronic Publishing, Uniersity of Michigan Press, ol. 2, no. (996). He is an Associate Editor of Operations Research, Management Science, Telecommunications Systems, etwork and Spatial Economics, and Computers and Operations Research. He is a Senior Member of IFORMS (Institute For Operations Research and Management Science), and on the Adisory Boards of a number of Information Technology start-up companies. eil J. Keon became a Member (M) of IEEE in eil has a Bachelor of Com. with oint honours in Economics and Finance (992) from McGill Uniersity, an M.Sc in Systems Science. (995) from the Uniersity of Ottawa, and an M.S.E (999) and a Ph.D. (2000) in Systems Engineering from the Uniersity of Pennsylania. He is an Assistant Professor in Information Technology and Operations Management at the Co School of Business, Southern Methodist Uniersity, Dallas TX, where he has been since He is also a member of IFORMS. G. Anandalingam (M 85, SM 90) has a B.A. (975) from Cambridge Uniersity in Electrical Sciences, and Ph.D. (98) in operations research with economics from Harard Uniersity. Anand is the Ralph J. Tyser Professor of Management Science at the Robert H. Smith School of Business, and the Institute for Systems Research at the Uniersity of Maryland where he has been since 200. From , he was the ational Center Professor (Engineering School), and Professor of Operations & Information Management (Wharton School) at the Uniersity of Pennsylania. He has published more than 70 papers on telecommunications and networking issues, and a book on Telecommunications etwork Design to be published by Kluwer Scientific Publishers.